Nadav M. Shnerb

Facilitation, competition, and vegetation patchiness: from scale free distribution to patterns.

A new technique for the modeling of perennial vegetation patchiness in the arid/semiarid climatic zone is suggested. Incorporating the stochasticity that affects life history of seedlings and the deterministic dynamics of soil moisture and biomass, this model is flexible enough to yield qualitatively different forms of spatial organization. In the facilitation-dominated regime, scale free distribution of patch sizes is observed, in correspondence with recent field studies. In the competition controlled case, on the other hand, power-law statistics is valid up to a cutoff, and an intrinsic length scale appears.

Extinction Rates for Fluctuation-Induced Metastabilities: A Real-Space WKB Approach

The extinction of a single species due to demographic stochasticity is analyzed. The discrete nature of the individual agents and the Poissonian noise related to the birth-death processes result in local extinction of a metastable population, as the system hits the absorbing state. The Fokker-Planck formulation of that problem fails to capture the statistics of large deviations from the metastable state, while approximations appropriate close to the absorbing state become, in general, invalid as the population becomes large. To connect these two regimes, a real space WKB method based on the master equation is presented, and is shown to yield an excellent approximation for the decay rate and the extreme events statistics all the way down to the absorbing state. The details of the underlying microscopic process, smeared out in a mean field treatment, are shown to be crucial for an exact determination of the extinction exponent. This general scheme is shown to reproduce the known results in the field, to yield new corollaries and to fit quite precisely the numerical solutions. Moreover it allows for systematic improvement via a series expansion where the small parameter is the inverse of the number of individuals in the metastable state.

Niche versus neutrality: a dynamical analysis.

Understanding the forces shaping ecological communities is of crucial importance for basic science and conservation. After 50 years in which ecological theory has focused on either stable communities driven by niche-based forces or nonstable “neutral” communities driven by demographic stochasticity, contemporary theories suggest that ecological communities are driven by the simultaneous effects of both types of mechanisms. Here we examine this paradigm using the longest available records for the dynamics of tropical trees and breeding birds. Applying a macroecological approach and fluctuation analysis techniques borrowed from statistical physics, we show that both stabilizing mechanisms and demographic stochasticity fail to play a dominant role in shaping assemblages over time. Rather, community dynamics in these two very different systems is predominantly driven by environmental stochasticity. Clearly, the current melding of niche and neutral theories cannot account for such dynamics. Our results highlight the need for a new theory of community dynamics integrating environmental stochasticity with weak stabilizing forces and suggest that such theory may better describe the dynamics of ecological communities than current neutral theories, deterministic niche-based theories, or recent hybrids.

Inverse melting and inverse freezing: a spin model.

Systems of highly degenerate ordered or frozen state may exhibit inverse melting (reversible crystallization upon heating) or inverse freezing (reversible glass transition upon heating). This phenomenon is reviewed, and a list of experimental demonstrations and theoretical models is presented. A simple spin model for inverse melting is introduced and solved analytically for infinite range, constant paramagnetic exchange interaction. The random exchange analogue of this model yields inverse freezing, as implied by the analytic solution based on the replica trick. The qualitative features of this system (generalized Blume-Capel spin model) are shown to resemble a large class of inverse melting phenomena. The appearance of inverse melting is related to an exact rescaling of one of the interaction parameters that measures the entropy of the system. For the case of almost degenerate spin states, perturbative expansion is presented, and the first three terms correspond to the empiric formula for the Flory-Huggins chi parameter in the theory of polymer melts. The possible microscopic origin of this chi parameter and the limitations of the Flory-Huggins theory where the state degeneracy is associated with the different conformations of a single polymer or with the spatial structures of two interacting molecules are discussed.

A neutral theory with environmental stochasticity explains static and dynamic properties of ecological communities.

Understanding the forces shaping ecological communities is crucial to basic science and conservation. Neutral theory has made considerable progress in explaining static properties of communities, like species abundance distributions (SADs), with a simple and generic model, but was criticised for making unrealistic predictions of fundamental dynamic patterns and for being sensitive to interspecific differences in fitness. Here, we show that a generalised neutral theory incorporating environmental stochasticity may resolve these limitations. We apply the theory to real data (the tropical forest of Barro Colorado Island) and demonstrate that it much better explains the properties of short-term population fluctuations and the decay of compositional similarity with time, while retaining the ability to explain SADs. Furthermore, the predictions are considerably more robust to interspecific fitness differences. Our results suggest that this integration of niches and stochasticity may serve as a minimalistic framework explaining fundamental static and dynamic characteristics of ecological communities.

Adaptation of autocatalytic fluctuations to diffusive noise.

Evolution of a system of diffusing and proliferating mortal reactants is analyzed in the presence of randomly moving catalysts. While the continuum description of the problem predicts reactant extinction as the average growth rate becomes negative, growth rate fluctuations induced by the discrete nature of the agents are shown to allow for an active phase, where reactants proliferate as their spatial configuration adapts to the fluctuations of the catalyst density. The model is explored by employing field theoretical techniques, numerical simulations, and strong coupling analysis. For d< or =2, the system is shown to exhibits an active phase at any growth rate, while for d>2 a kinetic phase transition is predicted. The applicability of this model as a prototype for a host of phenomena that exhibit self-organization is discussed.

LANGUAGE AND CODIFICATION DEPENDENCE OF LONG-RANGE CORRELATIONS IN TEXTS

We study the long-range correlations in various translations of the Bible by mapping them onto a one dimensional random walk model. We show that the exponent α, which characterizes the algebraic decay law of these correlations depends on both the language and the codification. Some considerations concerning the origin of these correlations are suggested in terms of which a qualitative explanation of this dependence is presented.

On Feynman’s approach to the foundations of gauge theory

In 1948, Feynman showed Dyson how the Lorentz force law and homogeneous Maxwell equations could be derived from commutation relations among Euclidean coordinates and velocities, without reference to an action or variational principle. When Dyson published the work in 1990, several authors noted that the derived equations have only Galilean symmetry and so are not actually the Maxwell theory. In particular, Hojman and Shepley proved that the existence of commutation relations is a strong assumption, sufficient to determine the corresponding action, which for Feynman’s derivation is of Newtonian form. In a recent paper, Tanimura generalized Feynman’s derivation to a Lorentz covariant form with scalar evolution parameters, and obtained an expression for the Lorentz force which appears to be consistent with relativistic kinematics and relates the force to the Maxwell field in the usual manner. However, Tanimura’s derivation does not lead to the usual Maxwell theory either, because the force equation depends o...

Temporal fluctuation scaling in populations and communities.

Taylors law, one of the most widely accepted generalizations in ecology, states that the variance of a population abundance time series scales as a power law of its mean. Here we reexamine this law and the empirical evidence presented in support of it. Specifically, we show that the exponent generally depends on the length of the time series, and its value reflects the combined effect of many underlying mechanisms. Moreover, sampling errors alone, when presented on a double logarithmic scale, are sufficient to produce an apparent power law. This raises questions regarding the usefulness of Taylors law for understanding ecological processes. As an alternative approach, we focus on short-term fluctuations and derive a generic null model for the variance-to-mean ratio in population time series from a demographic model that incorporates the combined effects of demographic and environmental stochasticity. After comparing the predictions of the proposed null model with the fluctuations observed in empirical data sets, we suggest an alternative expression for fluctuation scaling in population time series. Analyzing population fluctuations as we have proposed here may provide new applied (e.g., estimation of species persistence times) and theoretical (e.g., the neutral theory of biodiversity) insights that can be derived from more generally available short-term monitoring data.

Universal features of surname distribution in a subsample of a growing population

We examine the problem of family size statistics (the number of individuals carrying the same surname, or the same DNA sequence) in a given size subsample of an exponentially growing population. We approach the problem from two directions. In the first, we construct the family size distribution for the subsample from the stable distribution for the full population. This latter distribution is calculated for an arbitrary growth process in the limit of slow growth, and is seen to depend only on the average and variance of the number of children per individual, as well as the mutation rate. The distribution for the subsample is shifted left with respect to the original distribution, tending to eliminate the part of the original distribution reflecting the small families, and thus increasing the mean family size. From the subsample distribution, various bulk quantities such as the average family size and the percentage of singleton families are calculated. In the second approach, we study the past time development of these bulk quantities, deriving the statistics of the genealogical tree of the subsample. This approach reproduces that of the first when the current statistics of the subsample is considered. Surname statistics for the US in 1790 and 2000 and for Norway in 2008 are analyzed in the light of the theory and show satisfactory agreement, when the time-dependence of the growth rate is taken into account for the two contemporary data sets.